J an 2 00 8 Stability in the Stefan problem with surface tension ( I )

The Stefan problem is one of the best known parabolic two-phase free boundary problems. It is a simple model of phase transitions in liquid-solid systems. Let Ω⊂Rn denote a domain that contains a liquid and a solid separated by an interface Γ. As the melting or cooling take place the boundary moves and we are naturally led to a free boundary problem. The unknowns are the temperatures of the liquid and the solid denoted respectively by v and v− and the location of the interface Γ separating the two different phases. We shall assume that Ω=Tn−1× [−1,1] where Tn−1 stands for an (n−1)dimensional torus. Let us assume that the moving interface Γ(t) is a graph given by xn=ρ(t,x ′). Here ρ : [0,T ]×Tn−1→R is some smooth function such that ⋃ 0≤t≤T Γ(t)⊂Ω and T >0. Define the liquid/solid phase Ω±(t) by setting Ω±(t)= {